A combinatorial approach to the conditioning of a single entry in the stationary distribution for a Markov chain

Abstract

For an irreducible stochastic matrix T of order n, a certain condition number Κj(T) that measures the sensitivity of the j-th entry of the corresponding stationary distribution under perturbation of T is considered. A lower bound on Κj is produced in terms of the directed graph of T, and the case of equality is characterized in that lower bound. Also all of the directed graphs D are characterized such that Κj (T) is bounded from above as T ranges over the set of irreducible stochastic matrices having directed graph D. For those D for which Κj is bounded, a tight upper bound is given on Κj in terms of information contained in D.